Saturday, September 12, 2015

Sequence Numbers that Show Multiples of a Number

Can we adapt what we know about counting
sequences to create sequences that show multiples of numbers (other than one)?

Suppose we want to have a digital sequence
that shows multiples of 3 (or as some people would say “counts by
threes”). I could start with the
sequence number 998001, and its inverse 1/998001. I haven’t changed anything yet – so I know it
still counts by ones.

But if I do change it by multiplying the
fraction by 3 (3/998001) well, this is not the inverse of an integer – we need
to have a 1 on top of the fraction. That’s OK here, because 3/998001 simplifies
(the numerator and the denominator are both divisible by 3) and I get 1/332667.

We were counting by ones, and we multiplied
by three so now we are counting by threes (or multiples of three). And the 3 in the numerator conveniently
canceled with a factor of three in the denominator. So we are good to go with this example.

So the secret here is to know which numbers
will factor out and which won’t. Threes
will almost always factor out (unless you have to many threes to factor
out). Twos and fives will never factor
out (the sequence numbers we have use all end in with a 1 – so they can’t be
divisible by 2 or by 5).

Notice that they don’t all have the same
factors, so not every sequence number will work every time. But if you look hard you may find some that
do work for your specific situation.

999,998,000,001 has factors of 3, 7, 11,
13, and 37, so I can use it to find a sequence number that lists its term in 6
digit strings for multiples of 3, 7, 11, 13, and 37. So 3/999998000001, 7/999998000001, 11/999998000001,
13/999998000001, and 37/999998000001 can all be used – they will all simplify.

I’m not going to list all of the
possibilities, but I will give you some examples.

Suppose I wanted a sequence number that
would produce a sequence of terms that were all of the multiples of 123. The prime factors of 123 are 3 and 41. So I need to find a number in the table above
that has factors of 3 and 41. 9,999,800,001
will work, but so will 99,999,999,980,000,000,001 and
999,999,999,999,998,000,000,000,000,001.
The first will produce terms written in five digit strings, the second
in 10 digit strings, and the third in 15 digit strings. Since multiples of 123 will grow faster that
multiples of 1, I need to pick a string size that is appropriate (If you are
not sure just guess, check, and try again until you find what you need.). I’m going to try the middle one. If I don’t like the results, I can change it

This sequence number produces multiples
of 123, beginning with 0 * 123, and writing them in 10 digit strings.

Accurate to the 390th non-zero
term (390 * 123 = 47970), which is the limit of my computation.

This sequence is not in the OEIS.

We still have problem producing sequence
numbers that end up showing multiples of some numbers.

You may have noticed that all of the
sequence numbers that produce counting sequences end in a “1”. Any number that ends in 1 cannot be divisible
by 2 or 5. So I did not show you a
sequence number that shows multiples of 2 or 5 – yet.

I have developed two methods of dealing
with this issue – but you may not consider them to be valid solutions. For now just watch and learn. Then you will have time to think about these
methods and decide if you thing they are OK, or not OK.

Multiples of 2

2/998001 does not simplify. However, 2/998001 can be written as the sum
of two unit fractions 1/499001 and 1/498003497001. (Which brings up the issue of using more
than 1 fraction – is it fair?) So
let’s do it.

Accurate to 498th non-zero
term (996), then it skips 998, and it starts doing odd numbers. (Note that 998 is the only one, two, or three
digit even number that it skipped.)

Compare with OEIS sequence A005843.

Multiples of 5:

5/998002 does not simplify, but it can be
converted into three fractions that all have 1 as their numerator. I can add the results of each of these
fractions to get the results that I am looking for. This means it is not at type 1 sequence
number, but it is still mathematically kewl.
(“Kewl” is pronounced like “cool” and “kool”, but it does not refer to
the temperature, a childs drink, or a brand of cigarettes.)

The sequence only skips one multiple of 5
(1, 2, or 3 digit multiples of 5).

It also has an extra zero at the
beginning of the decimal expansion. It
sticks out like a sore thumb.

Written in three digit strings.

Accurate up to 990, the 198th
non-zero term.

Compare with OEIS sequence A008587.

So at this point we know how to find a
sequence number that will produce a decimal expansion that counts as high as
we want one to count.

And we can find a sequence number that
will show multiples of 3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 73,
79, 101, 137, 211, 239, 241, 271, 281, 757, 859 (and some larger primes), and
some combinations of these numbers. We
also know that we can find ways to deal with some of the numbers that we can’t
deal with using the method mentioned above.

So right now you know more about how to
find sequence numbers than … well, everybody that has not visited this site –
which is almost everybody.

About Me

This blog is about a special class of numbers that I call Sequence Numbers. I have been working on them for a few years,and just recently things came together. Phillip is helping me get this material posted.